All of the 5th grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$7.00$ each for teachers and $$3.50$ each for students, and the group paid $$56.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$28.00$ each for teachers and $$10.50$ each for students, and the group paid $$189.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+3.5y = 56}$ ${28x+10.5y = 189}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-28x-14y = -224}$ ${28x+10.5y = 189}$ Add the top and bottom equations together. $ -3.5y = -35 $ $ y = \dfrac{-35}{-3.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {7x+3.5y = 56}$ to find $x$ ${7x + 3.5}{(10)}{= 56}$ $7x+35 = 56$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {28x+10.5y = 189}$ and get the same answer for $x$ ${28x + 10.5}{(10)}{= 189}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.